\subsection{}
I have done a very crude estimation on Pauli exclusion on two-channels.  

\begin{itemize}
	\item Assume we have a simple BEC of close-channel molecules as the solution in isolation, and open-channel as simple free fermions.  Open-channel fermions have to occupy some levels over original $E_F$ because close-channel molecules occupy some within old Fermi-sea. This cost some energy. 
	\item Assume opne-channel is simple free fermions and simply occupy states up to $E_F$. Close-channel has to form molecules from $E_F$, which costs a bit more energy and reduce the binding energy.  
\end{itemize}
In both 2D and 3D, the previous case costs less energy then the later one.  In general, the later is in order of $E_b/E_F$ of the former($E_b$ is the bound energy of close-channel molecules and much larger than $E_F$, the open-channel fermi energy).  This supports my previous treatment.  My treatment is mostly according to the senario of the first one, which costs less energy. 

\subsection{Expansion of $B^\dagger{}^N$ state}
According to Combescot, it should be $B_0^\dagger{}^N+\sum_{i,j}(...\;)B_0^\dagger{}^{N-2}B_i^\dagger{}B_j^\dagger+\sum_{i,j,l,m}(...\;)B_0^\dagger{}^{N-4}B_i^\dagger{}B_j^\dagger{}B_l^\dagger{}B_m^\dagger$